Exponential stability for an infinite memory wave equation with frictional damping and logarithmic nonlinear terms
Qingqing Peng, Yikan Liu
TL;DR
The paper proves exponential stability for an infinite-memory wave equation with a logarithmic nonlinearity and frictional damping in a bounded domain with mixed boundary conditions. It constructs a weighted energy framework incorporating memory and a density function ρ(x), proves a lower energy bound, and establishes exponential decay under structural and initial-energy constraints using a contradiction argument, multiplier methods, and microlocal analysis. A special choice of ρ can eliminate the need for damping, showing memory alone can stabilize the system. The results extend viscoelastic-wave stability analyses to nonlinear logarithmic terms and acoustic boundaries, highlighting the interplay between memory, damping, and geometry.
Abstract
This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in $\mathbb R^d$ ($d\ge3$) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density $ρ(x)$ under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if $ρ(x)$ takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the system.